Wave Character of the Time Dependent Ginzburg Landau Equation and the Fluctuating Pair Propagator in Superconductors*)

Progress of Theoretical Physics, Oct 1971

The time dependent Ginzburg Landau equation and the fluctuation propagator of a superconductor in a static electromagnetic field are examined to the next order of the small parameters 1/εFτ and T/εF, where εF is the Fermi energy and τ the relaxation time. The coefficient of the time derivative of the order parameter becomes a complex number, whose imaginary part is of order of T/εF smaller than the real part. This term is shown to be important for Hall effect due to the fluctuation near Tc, which has not been expected in the usual approximation. The argeement cover the cases of arbitrary mean free paths near Tc and of arbitrary temperatures except T≪Tc.

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Wave Character of the Time Dependent Ginzburg Landau Equation and the Fluctuating Pair Propagator in Superconductors*)

Progress of Thc\.retical Physics Wave Character of the Time Dependent Ginzburg Landau Equation and the Fluctuating Pair Propagator in Superconductors*> Hiromichi EBISAWA 0 1 Hidetoshi FUKUYAMA 0 1 0 Department of Physics, University of Tokyo , Tokyo 1 > Part of Ph. D. Thesis submitted by H. Ebisawa to University of Tokyo , 1970 The time dependent Ginzburg Landau equation and the fluctuation propagator of a superconductor in a static electromagnetic field are examined to the next order of the small parameters 1/SF't" and TfsF, where sF is the Fermi energy and 't" the relaxation time. The coefficient of the time derivative of the order parameter becomes a complex number, whose imaginary part is of order of TlsF smaller than the real part. This term is shown to be important for Hall effect due to the fluctuation near Tc, which has not been expected in the usual approximation. The arguments cover the cases of arbitrary mean free paths near Tc and of arbitrary temperatures except T<f:.Tc. - Wave Character of the Time Dependent Ginzburg Landau Equation 1043 Integration over a momentum variable is transformed into the energy integral multiplied by the density of states, N, at the Fermi energy. By these approximations, we have with ID (q, w +. io) = - N_!_ [77- il-ow + Aq2]-\ f) . 1 r-?Jf (r, t) = - P2?JI' (r, t) - a?JI' (r, t)' fJt 2m A=nD 8T' a=~7J2mtt' r = 2~tt, where v and r are the Fermi velocity and relaxation time, respectively. Thus all coefficients are real positive numbers. In Eqs. ( 1 ) and ( 2 ), we neglect terms of the order of 1/sFr and T/sF, where SF is the Fermi energy. Using ID, Eq. ( 1 ), or the TDGL equation, Eq. ( 2 ), we can examine such dynamical properties as electrical resistivity and the ultrasonic attenuation. However as regards the Hall effect, Caroli and Maki3> obtained a vanishing contribution in the vortex state, and Tsuzuki and the present authors6> showed that the Hall conductivity does not have contributions from the AL process slightly above Tc. These van­ ishing results on Hall effects are intimately connected with the fact that Ao and A in Eq. ( 1 ) or Eq. ( 2 ) are real quantities. The reality of Ao and A are, however, true only if we neglect terms of the order of 1/sFr and T/sF in the derivation of ID. Thus we are in need to determine the fluctuation propagator ID and the TDGL equation to this order to examine some dynamical processes. These de­ terminations up to this order in the presence of external electric and magnetic fields are the purpose of the present article. We confine ourselves within the Born approximations and the ladder corrections to impurity scattering. That is, we work within (i) and (ii), but need not (iii) and (iv). In § 1 the derivation is given for the nearly free electron system to clarify the expansion parameters. As is shown the corrections, which is of order of T /sF, depend on the energy dependence of the density of states function near the Fermi energy. Then we consider in § 3 the cases of arbitrary Bloch electrons. In § 4 we briefly discuss a new effect due to these corrections. § 2. Fluctuation propagator and TDGL equation for a nearly free electron system For nearly free electrons, the structure of the propagator g) is very simple. The model Hamiltonian is ( 1 ) ( 2 ) ( 3 ) ( 4 ) where 1 . -=2nniu2N, r N is the density of states = 1 +X - - -lr + ... , 2sF 2 N= mkF. 2n2 u is the Fourier transform of U(r). · The branch of square root is such as Im <f?1R>O. From now on, the approximation, Re 1: = 0, is adopted. In the ladder approximation for the BCS coupling, the fluctuation propagator Is given by 1 i[i i[i Il(q, iw-;..) = - {3 0 0 where drdr' exp[iw-;..(r-r')J<T..P"t(q, r)P'( -q, r')), ( 13 ) ( 6 ) ( 7 ) ( 8 ) ( 9 ) ( 10 ) ( 11 ) ( 12 ) (14) Here IJR is an analytic continuation of II(q, iw-;..) from the upper plane of{)) and the bracket means the thermal average. Adopting the ladder approximation for impurity scattering, which is consistent with Eq. ( 8 ), one obtains IJR(q, w) =~fro dx tanh _!E_[flRR(x, x+ w) -flAR(x, x+w) 4nz -oo 2T +flAR(x-w, x) -flAA(x-()), x)], (15) where in the small q limit, where C(J2R(x')== j 1+ -x' -.SA( -x') e]i' x' i _-1+2e]i'+2r+···. (19) One can immediately see that the fourth term in Eq. (15) gives the same con­ tribution as the first. In the dirty limit Eq. (15) is calculated as =N- (1 +w- )\ sco dxtanhx 2 4eF -co 2T x {[{x+ 2 w+iDq ( 2 w )}-1 1+ 4c;F . _1] - (x+zo) 1 } + x+io ' which is valid to the linear order of w. Here we have performed the expansiOn of cp's with respect to 1/c;Fr and x/c;Fr-...JT/c;F (or WD/c;F) to the order following the one that leads to the ordinary ID. The expansion parameter with q2 is rDq2• Introducing the BCS cutoff, !xi <wD, in the second term, and using the relation l__JIR(O, 0) = l_-Nln 2rwD =ln_Z_, g g nT Teo one obtains JIR and then IIJR as follows: [I!JR(q,w)]-1=-N{ln ~o +w(~ +t:)-w(~) where ¢ 1s the di-gamma function and +_!_[_!_-ln_Z_- w(l_ + e:) + ¢ (l_)]}' 4c;F gN Teo 2 2 (= - iw + Dq 2 ( 47C T w ) 1 +4el-i', · (20) (21) (22) where X(p) is the Gor'kov function, B) given by X(p) = _8_1_ {n2 + l_[¢ (_!_) - ¢(_!_ + _!_P)} J 7( ( 3 ) p 8 2p 2 2 2 ' 1 p= 2nrT· So far we have neglected the effects of electric fields. For the derivation of the TDGL equation which includes a scalar potential of an electric field, careful treatments are necessary concerning the ordering of the operator 8/8r and the potential. Introducing an external field by In the presence of a magnetic field, q should be replaced by Q = q- 2eA.7> This result, Eq. (23), has new terms proportional to w, which is smaller than the ordinary terms by a factor T / cp. If Dq2/T<1, we get [.fDR(q, w)]-1= -N[ln T +J...q2 -iJ...0w+_!!!__-1-]~ Teo 4sFgN This equation is valid for cases with arbitrary values of mean free path, if we set 7( ( 3 )v2 J...- 48(nT)2 x(p), one has the equation for the order parameter Jt,9> to the linear order of V. The function P is defined by P(qh q2, iw}o) = - _!_ fP fP drdr' exp [iw}o (r- r')] {3 Jo Jo Jt(q, w) = g {JIR(q, w) Jt (q, w) + PR(qh q2, w) Jot Cq1) 2e V(q2, w)}, where x (T./Jft Cq1 + q2, r) 1Jf (- qh r) n (- q2, r')) n(q) = Jdr exp[ -iq·r]¢tt(r)¢t(r). In the dirty limit, a similar calculation to those for JIR yield PR(qh q2, w) for the limit, w___,.O (Appendix A). iN Dq2 ) PR(qt, q 2) = - 4nT ¢<1>(21 + 4nT 2 N [ 1 T ( 1 Dq1 ) ( 1 ) + 4cp gN-ln Teo-¢ 2+ 4nT +¢ 2 (24) (25) (26) (27) (28) (29) Wave Character of the Time Dependent Ginzburg Landau Equation 1047 -4rcTn~DQ2 fL2eV(r), 2sn +1DQ2 J2sn +1DQ2 }Jt(r,t) =O, where [ , ] is a commutator and Q=q-2eA. § 3. Fluctuation propagator in the case of Bloch electrons We assume an arbitrary Sk- k relation in a single band dispersion, restricting ourselves only to systems with the cubic symmetry. In this section we confine ourselves only in the dirty limit, and in the systems with a Fermi energy such that 1/sFr<1, T/sF<1 and (})D/sF<l. Equation (15) is still valid if one use the correct expressions for fl, i.e., for g. The self-energy is calculated as where N(x) is the density of states at cp+ x. Similarly one obtains gAR(x,x)= JdskN(sk) 1 . 1 . x-sk+Sp-z/2r -x-sk+sF-z/2r ~-ircN', where N' is the derivative of N with respect to x at x = 0. Using the relation and neglecting the quantity of the order of (})D/cp, one gets JIR, 2+3 (q, (})) = -(}). SIJJD dx tanhX - -d[gAR(x, x)- 1 -niu2]- 1 =0. 4rcz -IJ)n 2T dx (34) Another branch of g is expressed as gRR(x, x+(})) =~ GR(k, x)GA(k, -x-(})) k (32) (33) where the first term is expanded in terms of (J) as The k summation can be performed if one notes that one can expand the de­ nominator of G in terms of rx/"'.JrT. · ( :E GR(k,x)GA(k, -x) = SdeN(e) [eF-e+~z~-x 1+~z·~~N-')] k 2r 2r N 1 X [Sp-e-- z·~+x (1--z·__N_' )]2r 2r N 1 =2rcNr(1-2rx), :E GR(k, x)_LGA(k, -x-{f)) =- :E _J__GR(k, x)_J__GA(k, -x-(J)) k 8k/ k f)kp f)kp =- :E GR(k, x)GA(k, -x-(J)Y(_J__e(k)) .k f)~ =- :E(_]_e(k) rGR(k, x)2GA(k, -x)2 k f)kp 2 2 -2{)) (1--i_ N' ):E(_J__e(k)) GR(k, x)2GA(k, -x)3• 2r N k f)kp The first of Eq. (39) is evaluated as follows. - :E(-1__e(k) rGR(k, x)2GA(k, -x)2 k f)k,. · ( · N' )-. - 2 =- Sdea(e) [Sp-e+ dr -x 1+ dr N J X [Sp-S--z·-+x ( 1-- z·-N~~')] 2r 2r N 2 =4rcr3a(sF) (1+3irx), where (40) one obtains :E(_]__s(k) rGR(k, x)2GA(k, -x)S k f)k,. (37) (38) (39) (40) (41) gRR(x,x+U))=2nN{1+iU)r+2irx[1+iU)r(1- 1r ~)]}-2rcr3aq2• Thus we get where JI1+ 4 (q, 0)) =~ Jdx tanh ~[gRR(x, x + 0))-1 - niu2] 2nz 2T =N[1 +2iU)r (1-_i__ N' )] fdx tanh~ 1 2 4r N 2 T x + i( ' 1 ( = - - (- iU) +Dq2) [1 + 2iU)r (1 - _i_ N' ) ] , 4nT 4r N D=ar2/N. +0) -N'-[ -1 -<J; (-1+() +<J; (-1)' -lnT~J} . 2 N .gN 2 2 Te § 4. Discussion The same procedures of the integration over x as used In § 2 yield Again q should be replaced by Q = q- 2eA in the presence of a magnetic field. We have derived the fluctuation propagator and the TDGL equation for the order parameter and found the new terms proportional to 0) of order of TjsF. Though it is small it is important in some situation. One can show that there is a contribution to Hall conductivity due to fluctuation by use of the newly derived TDGL equation, Eq. (31), instead of Eq. ( 2 ). The arguments are restrict­ ed in the range near Te. The basic equations are h(r+is)( :t +2ie¢)Jt(r, t) =- [ 2~( -ir- 2:Ar+a]Jt(r, t), j(r, t) =- i!__(y -P' +4i~A)Jt(r', t)A(r, t) I , 2m c ~~ where e=2-ar. 7C (42) (43) (44) (45) (46) (47) (48) As we are concerned only with weak magnetic field cases, we apply the Wigner representation10> to treat the magnetic field for the system of fluctuating order parameters. Using the formal solution Jt (t), one gets the density matrix of the system as where p=(J(t)Jt(t)) =2-kBT- st dt' exp [( hr -co h(r- is) !}[ h(r+ ie) i~ext ) (t- t') J•1 + i!J{ext ) (t _ t') J h ' !J[ =-1'It2 +a, 2m 1t = p +2.!!.__ A , c !}[ext= 2eEx . The bracket means ensemble average with respect to stochastic variables, and 1 means the unit matrix. The factor in front of the integral in Eq. (50) is adjusted such that in the absence of external fields p may be consistent with GL free energy. Constructing the Liouville operator for the Wigner distribution function I corresponding to p from the equation of motion, one obtains fi/ = i (.£o + .£' + Lext) I+ 2kBT ' fit hr i.£ = - 2 ( -1-n2 +a) - .£ !!_ _}_, hr 2m /!. m fix l·.,Lr' -_ - e-2(J-)c ( 7Cfxi--7C yfi -) r2 finy finx ' i.£ext = 2eE_}_. finx fD= kBT (1/2m) 7t2 +a The equilibrium distribution function IS Following Kubo's derivation of the expression for the conductivity tensor to the linear order of H, one gets (50) (51) (52) (53) (54) Wave Character of the Time Dependent Ginzburg Landau Equation 1051 where the restriction of phase integral is used. Explicit calculations lead to The temperature dependence is thus found more singular than (J~x by Aslamazov­ Larkin.1) The full discussion of this large contribution to the Hall conductivity due to fluctuations will be given in a following paper, based on the microscopic theory, including contributions form the Maki process.11) A note is added concerning the difference of the new term in the fluctuation propagator or the TDGL equation form the corresponding one by Abrahams­ Tsuneto12) and Maki.18) Theirs include the q2 term as ( D+ 4_mi)q2. As is shown in Appendix B, g)R (q, w) is a real number for w= 0 in any approxi­ mation for the impurity scattering other than the Born and the ladder one as far as we work in the ladder approximation for the BCS coupling. In such approximations we may expect the corrections of order of 1/c.ur to the coefficient of w, which do not appear in the ladder approximation for the impurity scattering. The consistent treatments over 1: and the vertex corrections must be done. One more open problem now is the validity of neglecting the real part of 1:, which is closely related to the model potential due to impurities. Acknowledgements The authors thank Professor T. Tsuzuki, Professor K. Maki, Professor R. Kubo and Professor Y. W ada for useful discussions. They are grateful for financial support by the Research Institute for Fundamental Physics, Kyoto U niver­ sity. One of the authors (H.F.) is thankful to the Sakkokai Foundations for financial support. Appendix A Calculation of pR (q, q') In the ladder approximation Eq. (29) becomes (58) (59) X [1- niu2g(qh isn, isn) ]-1 [1- niu2g(q2, isn, - isn + iwA) ]-1 X [1- niu2g(q1 + q2, isn, isn- iwA) ]-1• (A·1) The analytically continued function of the summand is calculated similarly to fjBB', as 1 X(/-}11-+--(/-)2---i-r----Cq-1-+--q2-)-2/-3-k-F-2·-i-r1- -(c-p-/-+-c-p2-) 2 (<Pl + <P.'~ (<P.' + <P•) J + (q1 +kFq22)2 (terms obtam.ed by exchange of cp1 and cp/) }, 3 where cp~, cp2 or cp/ means the abbreviation of cp1B (x), cp2B (x) or cp1B' (x'), respec­ tively. Here q'2 is put to zero because the space charge, P2V, is vanishing. Expanding cp's in terms of T/sF and 1/sFr, one obtains, to the lowest order of {)), P R Cq1 q2) = ' 4 dx t a n2XTh-2-XsF- -x+-iD-1/-2·-q1-2 -x+-i-D-/2-1· -(q1-+-q2-)2 +_wN_Jdxtanh-x__d_[ 2 1 _1_(1+-i-D(q1+q2Y)] 4 2T dx w(1-x/2sF) x-iD/2·q12 -2 4sF ' (A·2) which leads to Eq. (30). Reality of ilJR(q, 0) By definition, we have IJR(q, 0) =- ~ fdx rdrexp[iq(r-r')] 4m J Appendix B where bar means the ensemble average over the random distribution of impuri­ ties. In Eq. (B ·1) we have put {)) = 0, for no singularity appears if (J)---)0. Impurity potential is real and the scattering is elastic. Thus JIR(q, 0) 1s real. [GR(r, r', x)]*=GA(r, r', x). D o w n l o a d e d f r o m h t t p : / / p t p . o x f o r d j o u r n a .l s o r g / b y g u e s t o n J u n e 2 1 , 2 0 1 6 1) L. G. Aslamazov and A. I. Larkin , Phys. Letters 26A ( 1968 ), 238 ; Fiz. Tverd. Tela 10 ( 1968 ), 1104 [ Soviet Phys .- Solid State 10 ( 1968 ), 875 ]. 2) H. Schmidt , z. Phys . 216 ( 1968 ) , 336 . A. Schmid , z. Phys . 215 ( 1968 ) , 210 . 3) C. Caroli and K. Maki , Phys. Rev . 164 ( 1967 ), 591 . 4) H. Ebisawa and H. Takayama , Prog. Theor. Phys . 42 ( 1969 ), 1481 . 5) W. Weller , Phys. Stat. Solidi 35 ( 1969 ) , 573 . 6) H. Fukuyama , H. Ebisawa and T. Tsuzuki , Phys. Letters 33A ( 1970 ), 187 . 7) K. Maki , Physics 1 ( 1964 ), 21 . 8) L. P. Gor 'kov , Soviet Phys.-JETP 10 ( 1960 ), 998 . 9) H. Takayama and H. Ebisawa , Prog. Theor. Phys . 44 ( 1970 ), 1450 . 10) R. Kubo , J. Phys. Soc. Japan 19 ( 1964 ), 2127 . 11) K. Maki , Prog. Theor. Phys . 39 ( 1968 ), 897 . 12) E. Abrahams and T. Tsuneto , Phys. Rev . 152 ( 1966 ), 416 . 13) K. Maki , Phys. Rev. Letters 23 ( 1969 ), 1223 .


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Hiromichi Ebisawa, Hidetoshi Fukuyama. Wave Character of the Time Dependent Ginzburg Landau Equation and the Fluctuating Pair Propagator in Superconductors*), Progress of Theoretical Physics, 1971, 1042-1053, DOI: 10.1143/PTP.46.1042